As analyzed, though the ADMM (REF ) per se can be easily derived, it is keen to solve the \(q_h\) -subproblem (REF ), i.e., the problem (REF ). Because demanding computation is required yet only a medium- or low-accuracy solution can be targeted, it is not attractive to consider first-order algorithms for this subproblem. It is thus interesting to investigate how much more complicated if a second-order algorithm is applied to the problem (REF ). In this and the next subsections, we will show that, counter-intuitively, the Newtonian system of (REF ) can be appropriately reformulated and relaxed so that its computation reduces to solving a simple positive definite linear system, and then the benchmark active-set Newton method in, e.g., [1]}, [2]}, can be applied very efficiently. Computation of the Newton step is comparable with, and usually less than, that of a single iteration of the gradient projection method with some backtracking line-search strategy, while the accuracy is much higher. This is a convincing example of deriving model-tailored efficient algorithms by taking full advantage of the structure of the model under discussion.
